Aaron Peterson, Stephen Taylor
Abstract.We consider a surfaceM immersed inR3 with induced metric g=ψδ2 whereδ2 is the two dimensional Euclidean metric. We then
con-struct a system of partial differential equations that constrainM to lift to a minimal surface via the Weierstrauss-Enneper representation, demand-ing the metric is of the above form. It is concluded that the associated surfaces connecting the prescribed minimal surface and its conjugate sur-face satisfy the system. Moreover, we find a non-trivial symmetry of the PDE which generates a one parameter family of surfaces isometric to a specified minimal surface.
M.S.C. 2000: 53A10, 30C55.
Key words: Minimal surfaces, univalent functions, Weierstrauss-Enneper represen-tations, PDE symmetry analysis.
1
Introduction
Given twoC2 functions u and v that satisfy Laplace’s equation, a complex valued
harmonic functionf is defined via the combination:f =u+iv. The Jacobian of such a function is given byJf =uxvy−uyvx. We will only consider harmonic functions that
are univalent (injective) with positive Jacobian onD ={z : |z| < 1}. On a simply
connected domain D ⊂ C, a harmonic mapping f has a canonical decomposition f =h+g where h andg are analytic inD, which is unique up to a constant. The dilatation ω of a harmonic map f is defined by ω ≡ g′/h′. The following theorem
provides the link between harmonic univalent functions and minimal surfaces:
Theorem 1. (Weierstrass-Enneper Representation). Every regular minimal sur-face locally has an isothermal parametric representation of the form
(1.1)
vanishing only at the poles (if any) ofqand having a zero of precise order2mwherever
q has a pole of order m. Conversely, each such pair of functions p and q analytic and meromorphic, respectively, in a simply connected domainD generate through the formulas (1.1) an isothermal parametric representation of a regular minimal surface.
∗
Balkan Journal of Geometry and Its Applications, Vol.13, No.2, 2008, pp. 80-85. c
We will use (1.1) in the following form:
Corollary1. For a harmonic functionf =h+g, define the analytic functionsh
andg by h=Rz
pdζ and g =−Rz
pq2dζ. Then the minimal surface representation
(1.1) becomes
See [6], [3] for a further introduction to harmonic mappings. For an introduction to minimal surfaces see [5].
2
The isometric condition
Since the Weierstrass-Enneper representation theorem requires thatMhas an isother-mal parametrization, we requireE=Gand F= 0, which implies
(2.7) φ2= 0, φ2= 0, E= 2|φ|2.
The first two equations are identically satisfied. Expanding the constraint onE, and using the identity
Im{g}=2g, we have the Cauchy Riemann and isometric conditions:
(2.9) 1hu−2hv = 0 1hv+2hu= 0
(2.10) 1gu−2gv = 0 1gv+2gu= 0
(2.11) 1h2u+2h2u+1g2u+2g2u−2
p
3
Symmetry analysis
We now proceed to calculate the symmetry group of (2.9)-(2.11). For an introduction to symmetry methods see [2], [7], and [8]. The infinitesimal generator of the symmetry group of the system is given by
(3.1) v=cu∂
u+cv∂v+c1h∂1h+c2h∂2h+c1g∂1g+c2g∂2g+cψ∂ψ.
Since the system is first order, we need only consider the first prolongation
pr(1)v=v+1hu∂1hu+1h
v∂
1hv+2h
u∂
2hu+2h
v∂
2hv
(3.2) +1gu∂1gu+1g
v∂
1gv +2g
u∂
2gu+2g
v∂
2gv,
where theci are functions ofu, v, ψ,
1h,2h,1g, and2g. Applying the first prolongation
to the three PDE determines a system equations for the ci. Integrating this system
yields ten symmetries of (2.9)-(2.11).
v1=∂u, v2=∂v, v3=∂1h, v4=∂2h, v5=∂1g, v6=∂2g,
v7=−v∂u+u∂v, v8=−2h∂1h+1h∂2h, v9=−2g∂1g+1g∂2g,
v10=u∂u+v∂v+1h∂1h+2h∂2h+1g∂1g+2g∂2g.
Exponentiating these infinitesimal vector fields gives the symmetry transformations:
(3.3) h→h(z−s) g→g(z−s),
(3.4) h→h(z−is) g→g(z−is),
(3.5) h→h+s g→g,
(3.6) h→h+is g→g,
(3.7) h→h g→g+s,
(3.8) h→h g→g+is,
(3.9) h→h(eisz) g→g(eisz),
(3.10) h→eish g→g,
(3.11) h→h g→eisg,
(3.12) h→esh(e−sz) g→esg(e−sz).
In [1] the minimal symmetry group for the real minimal surface equation
(3.13) uxx(1 +u2y) +uyy(1 +u2x)−2uxuyuxy= 0,
was calculated. Many of the translational symmetries and anesf(e−sx, e−sy)
4
Discussion and applications
Consider the transformationh→eiθh, g →eiθg which preserves the metric E =
|h′|2+|g′|2. When θ = 0, this is simply a minimal surface specified by defining ψ.
Whenθ = π/2 we get the conjugate surface. Thus all intermediate surfaces, called associated surfaces, are isometric. Since all minimal surfaces can be constructed from parts of a helicoid and catenoid [4], the following examples are of interest. First, we draw attention to the catenoid, given byψ = cosh(v)2. It’s conjugate surface is the helicoid and the associated surfaces between the two are plotted overDin Figure 1.
Since all of the associated surfaces are isometric, geometrically they are equivalent. However, note the catenoid has topologyS1×Rwhere as the helicoid has topology R2.
Figure 1: Helicoid to Catenoid Transformation.
f be the harmonic mapping f =h+g where
(4.1) h=1
2
µ1
2log
·1 +z
1−z
¸
+ z
1−z2
¶
,
and
(4.2) g= 1
2
µ
1 2log
·
1 +z
1−z
¸
− z 1−z2
¶
,
which lifts to the catenoid. We make the transformation in equation (3.12) by letting
h→esh(e−sz) andg→esg(e−sz). Figure 2 gives several plots of this transformation
for varioussvalues. The topology of the half catenoid isR2for allsup to some value
between (0.3,0.4) where it changes to a punctured cylinder. Note ass→ ∞that the minimal surfaces eventually degenerate to a line, in a manner peculiarly similar to neckpinch singularities of the Ricci Flow.
Figure 2: Symmetry (3.12) fors={−1.2,−0.5,0,0.3,0.4,0.5,1,1.5,3}.
When (3.12) is applied to the helicoid, we find that the number of rotations of the helicoid about its axis are scaled. Thus we have:
Theorem 2. Let S be the helicoid over Dparameterized isothermally by
x= (sinhusinv,sinhucosv,−v). For helicoids S1 given by u∈(0,2π),v ∈(v0, v1),
It would be interesting to generalize the symmetry methods of this paper to higher dimensional Riemannian or Lorentizian manifolds. One would need a generalized Weierstrauss-Enneper. Moreover, we believe there are potential topological theorems coming from symmetry (3.12). For instance, if one calculates the one parameter fam-ily of minimal surfaces given by symmetry (3.12) and a simply connected minimal surface, does the topology always change from the plane toS1×R1 or some variant
thereof?
Acknowledgements. The second author would like to acknowledge NSF grant PHY-0502218.
References
[1] N. Bˆıl˘a,Lie Groups Applications to Minimal Surfaces PDE, DGDS 1 (1999), 1-9. [2] B. J. Cantwell,Introduction to Symmetry Analysis, Cambridge University, 2002. [3] J. Clunie, T. Sheil-Small,Harmonic univalent functions, Annales Academiæ
Sci-entiarum Fennicæ 9 (1985), 3-25.
[4] T. H. Colding, William P. Minicozzi II, Shapes of embedded minimal surfaces, PNAS 103 (2006), 11103-11105.
[5] U. Dierkes, S. Hildebrandt, A. Kuster, O. Wohlrab, Minimal Surfaces I, II, Springer-Verlag, 1992.
[6] P. Duren,Harmonic Mappings in the Plane, Cambridge University, 2004. [7] P. J. Olver,Applications of Lie Groups to Differential Equations, Springer-Verlag,
second edition, 1993.
[8] H. Stephani, Differential Equations: Their Solution Using Symmetries, Cam-bridge University, 1989.
Authors’ addresses:
Aaron Peterson
Dept of Mathematics, Brigham Young University, Provo, UT, 84602 USA.
E-mail: [email protected]
Stephen Taylor
Dept of Mathematics, SUNY at Stony Brook, Stony Brook, New York, 11794 USA.
